Question

Find all equilibrium points of the autonomous system. $$ \begin{aligned} &x^{\prime}=-x+x y \\ &y^{\prime}=y-x y \end{aligned} $$

Short answer

Answer: The equilibrium points for the given autonomous system are all the points on the x-axis, including the origin (0,0).

Step-by-step solution

Set the derivatives equal to zero

Write down the derivatives given in the problem, and set them equal to zero. $$ \begin{aligned} x' = -x + xy &= 0 \\ y' = y - xy &= 0 \end{aligned} $$

Solve for x in the first equation

Determine the value(s) of \(x\) when the first equation is equal to zero: $$ \begin{aligned} -x + xy &= 0 \\ x(1-y) &= x \end{aligned} $$

Break down cases for the first equation

We broke the equation down in this way because either \(x=0\) or \(1-y=1\): Case 1: \(x=0\) Case 2: \(1-y=1\)

Solve for x and y in each case

Find the equilibrium point(s) for each case: Case 1: If \(x=0\), then \(y'=0=y-0\cdot y\) which leads to \(y=0\). So, the equilibrium point for this case is \((0,0)\). Case 2: If \(1-y=1\), then \(y=0\). So, we get \(x'=0=-x+x\cdot 0\), which implies \(x\) can be any value. Thus, there are infinitely many equilibrium points of the form \((x,0)\).

Combine all the equilibrium points

Our equilibrium points found were: 1. \((0, 0)\) 2. Any point of the form \((x, 0)\) (That is, any point on the x-axis) The equilibrium points of the autonomous system are all the points on the x-axis, including the origin \((0,0)\).

Key concepts

Autonomous Differential Equations

An autonomous differential equation is a type of differential equation in which the rate of change of a variable is a function of the variable itself but not explicitly dependent on the independent variable, often time. Think of it as a self-driven system where the current state solely determines the next state.

The general form is given by \( \frac{dx}{dt} = f(x) \) for a single-variable system, where \( x \) is the dependent variable, \( t \) is the independent variable, and \( f \) is a function that describes the rate at which \( x \) changes over time.

In the given exercise, both \( x' = -x + xy \) and \( y' = y - xy \) are autonomous because neither side of the equations involves the independent variable, \( t \). This implies that the system's behavior can be entirely determined by its current state, without additional information about how that state was reached.

Understanding the behavior of such systems is crucial in fields like biology, engineering, and physics, where systems often naturally progress without external time-based input.

Solving Equilibrium Point Equations

Equilibrium points, also known as fixed points or critical points, in a system of autonomous differential equations are the points at which the system is at rest, or unchanging over time. In mathematical terms, these are the points where the derivatives \( x' \) and \( y' \) equal zero.

To find these points, we set the derivatives of the system equal to zero and solve for the variables. This often involves solving simultaneous equations, which can sometimes be simplified by considering individual variables in isolation, as seen in the provided step by step solution.

In the provided example, the solution is found by examining cases where either term of the product \( x(1-y) \) is zero, which arises naturally from the zero-product property (if \( ab = 0 \) then either \( a = 0 \) or \( b = 0 \)). This methodical approach to solving equilibrium point equations ensures that all potential fixed points are identified, allowing us to understand the comprehensive behavior of the system at rest.

Recognizing that \( x = 0 \) or \( y = 0 \) leads to tangible interpretations in the context of the system, such as a population reaching a stable state, an object ceasing movement, or a chemical reaction becoming balanced.

Phase Plane Analysis

Phase plane analysis is a graphical method used to study the behavior of a system of two first-order autonomous differential equations. It involves plotting the state variables against each other on a plane—often termed the 'phase plane'—to visualize the trajectory of the system over time.

Each point in this plane represents a possible state of the system, with the phase paths or trajectories showing how the system evolves from one state to another. This analysis is instrumental in understanding the stability of equilibrium points and observing patterns such as oscillations or convergence to steady states.

In our example, plotting the equilibrium points on the phase plane would show us that \( (0, 0) \) is a single equilibrium point, and every point on the x-axis forms a line of equilibrium points. Analyzing the system’s behavior around these points, such as \(x' \) and \(y' \) changing with small deviations from equilibrium, can reveal if the system returns to equilibrium (a stable point) or moves away from it (an unstable point).

This form of analysis provides important insights, especially when dealing with systems that model real-world phenomena, like predator-prey relationships or electrical circuits where visualizing flow and stability can be far more intuitive and enlightening than dealing with equations alone.